Transcript Analytic Solution of Non-Axisymmetric Isothermal Dendrites

Analytic Solution of
Non-Axisymmetric Isothermal Dendrites
G.B. McFadden and S.R. Coriell, NIST
and
R.F. Sekerka, CMU
•Introduction
•Ivantsov solution
•Horvay-Cahn 2-fold solution
•Small-amplitude 4-fold solution
•Estimate of shape parameter
•Summary
NASA Microgravity Research Program, NSF DMR
Dendritic Growth
Peclet number:
Ivantsov solution [1947]:
Stefan number:
Experimental Check of Ivantsov Relation
M.E. Glicksman, M.B. Koss, J.C. LaCombe, et al.
There is a systematic 10% - 15% deviation.
Experimental Check of Ivantsov Relation
Possible reasons for deviation:
•Proximity of sidearms or other dendrites (especially at low T)
•Convection driven by density change on solidification
•Residual natural convection in g
•Container size effects
•Non-axisymmetric deviations from Ivantsov solution
“… the diffusion field described by [the Ivantsov solution] is
based on a dendrite tip which is a parabolic body of revolution,
which is true only near the tip itself.” [Glicksman et al. (1995)]
Non-Axisymmetric Needle Crystals
Idea: Compute correction to
Ivantsov relation S = P eP E1(P) due
to 4-fold deviation from a parabola
of revolution.
Key ingredients:
• Glicksman et al. have measured the deviation S - P eP E1(P)
• LaCombe et al. have also measured the shape deviation [1995].
• Horvay & Cahn [1961] found an exact needle crystal solution with
2-fold symmetry, exhibiting an amplitude-dependent deviation in S
- P eP E1(P) [but wrong sign to account for 4-fold data …]
Non-Axisymmetric Needle Crystals
•Unfortunately, there is no exact generalization of the HorvayCahn 2-fold solution to the 4-fold case.
•Instead, we perform an expansion for the 4-fold correction, valid
for small-amplitude perturbations to a parabola of revolution.
•Horvay-Cahn solution is written in an ellipsoidal coordinate
system. We transform the solution to paraboloidal coordinates, and
expand for small eccentricity to find the expansion for a 2-fold
solution in paraboloidal coordinates.
•We then generalize the 2-fold solution to the n-fold case (n = 3,4)
in paraboloidal coordinates .
Steady-State Isothermal Model of Dendritic Growth
Temperature T in the liquid:
Note: T/z is a solution if T is.
 2 T + V  T/ z = 0
Conservation of energy:
Melting temperature:
-LV vn = k T/n
T = TM
Far-field boundary condition (bath temperture):
T  T = TM - T
 = thermal diffusivity
LV = latent heat per unit volume
V = dendrite growth velocity
k = thermal conductivity
Characteristic scales: choose T for (T – TM) and 2/V for length.
Ivantsov Solution [1947] (axisymmetric)
Parabolic coordinates [, , ] (moving system) :
Solid-liquid interface:
Conservation of energy:
Temperature field:
Horvay-Cahn Solution [1961] (2-fold)
Paraboloids with elliptical cross-section:
Here  is the independent variable, and b ≠ 0 generates an elliptical
cross section.
Solid-liquid interface is  = P, temperature field is T = T():
Conservation of energy:
For b = 0, the axisymmetric Ivantsov solution is recovered.
Expansion of Horvay-Cahn Solution
Procedure:
•Set b = P 
•Re-express Horvay-Cahn solution in parabolic coordinates
•Expand in powers of  for fixed value of P
Find the thermal field T(,,,), interface shape  = f(,,), and
Stefan number S() as functions of  through 2nd order
Expansion of Horvay-Cahn Solution
At leading order, we recover the Ivantsov solution:
At first order:
S(1) vanishes by symmetry:   -  corresponds to a rotation,    + /2
The solution has 2-fold symmetry in .
Expansion of Horvay-Cahn Solution
At 2nd order:
where:
2nd order
exact
P = 0.01
Expansion of n-fold Solution
Goal: Find correction S(2) for a solution with n-fold symmetry
where the leading order solution is the Ivantsov solution as before,
and the first order solution is given by
Expansion of 4-fold Solution
Key points:
•Fix the tip at z = P/2
•Fix the (average) radius of curvature
•Employ two more diffusion solutions: “anti-derivatives” (method of characteristics)
Expression for S(2)
A symbolic calculation gives the exact result:
Comparison with Shape Measurements
In cylindrical coordinates, our dimensional result is:
LaCombe et al. [1995] fit SCN tip shapes using:
For P  0.004, they find
Q()  –0.004 cos 4:
Comparison of shapes gives   –0.008, and
evaluating S(2) for P = 0.004 and  = -0.008 then gives
4-Fold Tip Shape
For P = 0.004 and  = -0.008:
Huang & Glicksman [1981]
Estimate for Shape Parameter
Surface tension anisotropy (n) (cubic crystal):
n = (nx,ny,nz) is the unit normal of the crystal-melt interface.
For SCN, 4 = 0.0055  0.0015 [Glicksman et al. (1986)].
For small anisotropy, the equilibrium shape is geometrically
similar to a polar plot of the surface free energy, and we have
Estimate for Shape Parameter
Idea: Dendrite tip is geometrically-similar to the [100]-portion
of the equilibrium shape.
For small 4 and r/z ¿ 1, the equilibrium shape is:
Our expansion for the dendrite shape:
From the SCN anisotropy measurement:
From the tip shape measurement:
Summary
• Glicksman et al. observe a 10% - 15% discrepancy in the Ivantsov
relation for SCN over the range 0.5 K < T < 1.0 K
• Horvay-Cahn exact 2-fold solution gives an amplitude-dependent
correction to the Ivantsov relation
• An approximate 4-fold solution can be obtained to second order in ,
with S = S(0) + 2 S(2)/2 + ...
• LaCombe et al. measure a shape factor   -0.008 for P  0.004
• Using  = 0.008 gives S/S(0) - 1 = 0.09
• Assuming the dendrite tip is similar to the [001] portion of the
anisotropic equilibrium shape gives  = - 0.011  0.003
References
• M.E.
Glicksman and S.P. Marsh, “The Dendrite,” in Handbook of Crystal
Growth, ed. D.T.J. Hurle, (Elsevier Science Publishers B.V., Amsterdam, 1993),
Vol. 1b, p. 1077.
• M.E. Glicksman, M.B. Koss, L.T. Bushnell, J.C. LaCombe, and E.A. Winsa,
ISIJ International 35 (1995) 604.
•S.-C. Huang and M.E. Glicksman, Fundamentals of dendritic solidification – I.
Steady-state tip growth, Acta Metall. 29 (1981) 701-715.
•J.C. LaCombe, M.B. Koss, V.E. Fradkov, and M.E. Glicksman, Threedimensional dendrite-tip morphology, Phys, Rev. E 52 (1995) 2778-2786.
• G.B. McFadden, S.R. Coriell, and R.F. Sekerka, Analytic solution for a nonaxisymmetric isothermal dendrite, J. Crystal Growth 208 (2000) 726-745.
•G.B. McFadden, S.R. Coriell, and R.F. Sekerka, Effect of surface free energy
anisotropy on dendrite tip shape, Acta Mater. 48 (2000) 3177-3181.
Material Properties of SCN